Dear Prathmesh Joshi,

2fold axes are known to intersect with different axes but not at any any angle in crystallography. Where does Your question come from and where does it relate to?

Regards,

Wolfgang

As Wolfgang already pointed out: in (classic) crystallography two-fold axis can either have 90°, 60°, 30°, 45° and ~35° to other crystallographic axes.

cf. slides 14 and 22-28 in the presentation (see link)

But maybe you have something different in mind?

Conference Paper Symmetry: theoretical concept versus practical meaning

Point symmetry operations are subject to strict conditions, namely all of them must satisfy the criteria of a mathematical group. Moreover, if these symmetry operations must be compatible with translations which is the case in crystallographic symmetry, only (one-,) two-, three-, four- and sixfold rotation symmetry can occur. In this case only 32 possible cases exist in 3D which are described in the above presentation by Gert Nolze.

This finite number of cases seriously limits the number of possible orientations of e.g. two fold axis.  

Dear Prathmesh Joshi,

                                                first of all, your question seems to be a little bit ambiguous. The "context of X-ray crystallography" belongs to the more general context  of the "crystallography" which , in turn, belongs to the context of the "crystalline state". 

Secondly, a twofold symmetry axis can belong to a single molecule (CO2, for instance) and can make any angle with another axis ( twofold, threefold, fivefold. etc...) of another molecule in the liquid or gaseous, or any amorphous  state. This represents the "infinite" degree of freedom of the disordered states of the matter.

But, when a population of atoms, molecules or ions are confined ( in order to reach the minimum of their potential energy) in a limited space to make bonds among them, they assume an ordered and periodic disposition  ( in two or three dimensions) This is a crystal. . Accordingly, the symmetry elements (axes, mirror planes, inversion centres) do repeat in the space (2D, 3D) respecting this periodicity It is very easy to show that the consequent periodicity imposes sharp constraints :

- on the number of symmetry axes ( 2, 3, 4, 6)

- on the mutual position of the symmetry axes, among themselves and with respect to the symmetry planes

- on the mutual position of the symmetry planes among themselves.

All that has been well described by Gert Nolze, and correctly confirmed by Gervais Chapuis.  Historically, the problem of  the crystalline symmetry was posed and solved in a short time period ( from the end of the XVIII to the half of the XIX century, i.e. from René Juste Hauy to Auguste Bravais), more or less one century before the beginning of the X-ray Crystallography (Max von Laue).

Many thanks for your kind attention. I' lI can send you the geometrical demonstration of the just mentioned limitation imposed by the crystalline state, if you are interested in, of course.

All the best

Dino